Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5211343 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1316 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(2,6)(3,4)(9,10,13,16)(11,14,12,15)(18,22)(19,25)(20,28)(30,32)(33,34)(35,40)(36,41)(37,45)(38,39)(42,47)(43,44)(46,48);(1,2,5,6)(3,7,4,8)(18,22)(20,28)(23,26)(27,31)(34,39)(36,40)(42,45)(44,48);(1,3,5,4)(2,7,6,8)(9,11,13,12)(10,14,16,15)(17,18)(19,26)(20,24)(21,22)(23,25)(27,32)(28,29)(30,31)(33,35)(34,40)(36,39)(37,46)(38,41)(42,44)(43,47)(45,48);(1,4,5,3)(2,7,6,8)(9,10)(11,14)(12,15)(13,16)(17,19,21,25)(18,23,22,26)(20,27,28,31)(24,30,29,32)(33,36,38,40)(34,41,39,35)(37,44,47,48)(42,46,45,43);(1,2)(3,8)(4,7)(5,6)(9,12,13,11)(10,15,16,14)(17,20,21,28)(18,24,22,29)(19,27,25,31)(23,30,26,32)(33,37,38,47)(34,42,39,45)(35,43,41,46)(36,44,40,48)Polycyclic presentation:
2^9|a^2=gi,b^2=gh,c^2=hi,d^2=fh,e^2=fi,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5362643);